Propriétés des matrices ``bien localisées'' près de leur diagonale et quelques applications. (Properties of matrices ``well localized'' near the diagonal and some applications)
type: Item
instance of: scholarly article
title: Propriétés des matrices ``bien localisées'' près de leur diagonale et quelques applications. (Properties of matrices ``well localized'' near the diagonal and some applications)
zbMATH Open document ID: 0722.15004
DOI: 10.1016/S0294-1449(16)30287-6
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1990-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78234
review text: Let (T,d) be a discrete metric space with the property that for every \(\epsilon >0\sum_{t\in T}\exp [-\epsilon d(s,t)]\) is bounded as s varies over T. For any positive real number \(\gamma\), let \({\mathcal E}_{\gamma}\) be the set of all matrices \((A_{s,t})_{s,t\in T}\) such that for all \(\gamma '<\gamma| A_{s,t}| \exp [\gamma 'd(s,t)]\) is bounded as s and t vary over T, i.e., the entries in A decay exponentially away from the diagonal. Such matrices define bounded operators on \(l_ 2(T).\) The author proves that \({\mathcal E}_{\gamma}\) is an algebra, and that if \(A\in {\mathcal E}_{\gamma}\) is invertible as an operator on \(l_ 2(T)\), then \(A^{-1}\in {\mathcal E}_{\gamma '}\) for some \(\gamma '\). He also proves a ``window'' lemma that shows that changes in the entries in one part of a matrix in \({\mathcal E}_{\gamma}\) do not affect the entries in \(A^{-1}\) that are located far away from the changed entries. More precisely, if \(\Omega\subset T\) and A and B are two invertible matrices in \({\mathcal E}_{\gamma}\) such that \(| A_{s,t}-B_{s,t}| \exp \{\gamma [d(s,\Omega)+d(t,\Omega)]\}\) is bounded as s and t vary over T, then for some positive \(\gamma '\), \(| A^{-1}_{s,t}-B^{- 1}_{s,t}| \exp \{\gamma '[d(s,\Omega)+d(t,\Omega)]\}\) is also bounded as s and t vary over T. Now assume (T,d) is a discrete metric space with the property that for some N \(\sum_{t\in D}| 1+d(s,t)|^{-N}\) is bounded as s varies over T. For any \(\alpha >N\), let \(Q_{\alpha}\) be the set of all matrices \((A_{s,t})_{s,t\in T}\) such that \(| A_{s,t}[1+d(s,t)]^{\alpha}|\) is bounded as s and t vary over T. The author proves the same type of results for \(Q_{\alpha}\) as for \({\mathcal E}_{\gamma}.\) Two applications concerning wavelets are given.
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A30
zbMATH DE Number: 4189914
zbMATH Keywords: symbolic calculus
zbMATH Keywords: inversion of large matrices
zbMATH Keywords: approximation of inverses
zbMATH Keywords: asymptotic estimations of wavelets
zbMATH Keywords: discrete metric space
zbMATH Keywords: bounded operators
zbMATH Keywords: algebra
author: Stéphane Jaffard
reviewed by: Robert L. Ellis
MaRDI profile type: Publication
cites work: Q3971549
cites work: Bases d'ondelettes dans des ouverts de \({\mathbb{R}}^ n\). (Wavelet bases in open sets of \({\mathbb{R}}^ n)\)
cites work: Ondelettes et bases hilbertiennes. (Wavelets and Hilbert bases)
cites work: Q3995454
cites work: Q4185363
cites work: Q3725017
OpenAlex ID: W2572784842
Wikidata QID: Q105583808
description: scientific article; zbMATH DE number 4189914